110.201 Linear Algebra
1st Quiz Solution
February 15, 2005
Problem 1
Given the following system of equations:
3
x
+ 2
y
-
5
z
= 1
4
x
-
y
+
z
= 0
x
-
z
= 2
find all solutions using Gauss-Jordan elimination procedure. Is this an example
of consistent system? Why?
Solution:
We use Gauss-Jordan:
3
2
-
5
4
-
1
1
1
0
-
1
1
0
2
We rewrite the matrix as :
1
0
-
1
3
2
-
5
4
-
1
1
2
1
0
-
3
×
(I)
-
4
×
(I)
then:
1
0
-
1
0
2
-
2
0
-
1
5
2
-
5
-
8
÷
2
then:
1
1
0
-
1
0
1
-
1
0
-
1
5
2
-
5
2
-
8
+(II)
then:
1
0
-
1
0
1
-
1
0
0
4
2
-
5
2
-
21
2
÷
4
then:
1
0
-
1
0
1
-
1
0
0
1
2
-
5
2
-
21
8
+(III)
+(III)
then we have:
1
0
0
0
1
0
0
0
1
-
5
8
-
41
8
-
21
8
Therefore, we have the answer :
x
=
-
5
8
y
=
-
41
8
z
=
-
21
8
Problem 2
Find the rank of the following matrix
-
1
3
8
-
2
1
-
1
3
9
-
1
3
1
-
3
-
9
1
-
3
0
0
0
0
2
Solution:
We find reduced row-echelon form of this matrix:
-
1
3
8
-
2
1
-
1
3
9
-
1
3
1
-
3
-
9
1
-
3
0
0
0
0
2
×
(
-
1)
+(II)
÷
2
2
then:
1
-
3
-
8
2
-
1
-
1
3
9
-
1
3
0
0
0
0
0
0
0
0
0
1
+(I)
then:
1
-
3
-
8
